[Fwd: pair sums applied to trignometry sums]
From: Roger Bagula (tftn_at_earthlink.net)
Date: 11/28/04
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Date: Sun, 28 Nov 2004 19:26:02 GMT
-------- Original Message --------
Subject: pair sums applied to trignometry sums
Date: Sun, 28 Nov 2004 11:23:55 -0800
From: Roger Bagula <tftn@earthlink.net>
Reply-To: tftn@earthlink.net
Organization: tftn/bmftg
To: chaos theory <caostheory@yahoogroups.com>, true number theory
<truenumber@yahoogroups.com>, number theory
<numbertheory@yahoogroups.com>, MathematicaUsers@yahoogroups.com,
fractals <fractals@groups.msn.com>
I had used the mechanism with Bailey type of sequences
and their sums in the work on b normalness in iteratives functions.
It occurred to me that by adding the variable x , I could get
functiond that used the nonlinear Cantor pair {1/(n+1),n/(n+1)}
to split the sine and the cosine down the middle.
The result is entirely new trignometric sum functions that converge very
well.
(* pair sums applied to trignometry sums: {1/(n+1),n/(n+1)} modulo 2
switched sums*)
(* these sums break the trignometry of a circle into four functions
instead of two*)
(* these are subharmonic functions of a nonlinear Rational Cantor type*)
fs[x_,n_]:=
If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^(n)*
x^(2*n+1)/((n+1)*(2*n+1)!)]
gs[x_,n_]:=
If[Mod[n,2]==1,(-1)^n*x^(2*n+1)/((n+1)*(2*n+1)!),(-1)^n*n*
x^(2*n+1)/((n+1)*(2*n+1)!)]
fc[x_,n_]:=
If[Mod[n,2]==1,(-1)^(n)*n*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)*
x^(2*n)/((n+1)*(2*n)!)]
gc[x_,n_]:=
If[Mod[n,2]==1,(-1)^(n)*x^(2*n)/((n+1)*(2*n)!),(-1)^(n)*n*
x^(2*n)/((n+1)*(2*n)!)]
digits=100;
fsin[x_]:=N[Sum[fs[x,n],{n,0,digits}]];
gsin[x_]:=N[Sum[gs[x,n],{n,0,digits}]]
fcos[x_]:=N[Sum[fc[x,n],{n,0,digits}]]
gcos[x_]:=N[Sum[gc[x,n],{n,0,digits}]]
Plot[fsin[x],{x,-Pi,Pi}]
Plot[fsin[x],{x,-Pi,Pi}]
Plot[gsin[x],{x,-Pi,Pi}]
Plot[fcos[x],{x,-Pi,Pi}]
Plot[gcos[x],{x,-Pi,Pi}]
Plot[(fsin[x]+gsin[x]),{x,-Pi,Pi},PlotRange->All]
Plot[(fcos[x]+gcos[x]),{x,-Pi,Pi}]
ParametricPlot[{fsin[x],gsin[x]},{x,-Pi,Pi}]
ParametricPlot[{fcos[x],gcos[x]},{x,-Pi,Pi}]
ParametricPlot[{fsin[x],fcos[x]},{x,-Pi,Pi}]
ParametricPlot[{gsin[x],gcos[x]},{x,-Pi,Pi}]
ParametricPlot[{fsin[x]+gsin[x],fcos[x]+gcos[x]},{x,-Pi,Pi},PlotRange->All]
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn
-- Respectfully, Roger L. Bagula tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 : alternative email: rlbtftn@netscape.net URL : http://home.earthlink.net/~tftn
- Previous message: Roger Bagula: "Re: The Fractal Challenge"
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